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Autonomous Categories in Which a = (Extended Abstract) Autonomous categories in which A ∼= A∗ (extended abstract) Peter Selinger Dalhousie University Abstract Recently, there has been some interest in autonomous categories (such as compact closed categories) in which the objects are self-dual, in the sense that A =∼ A∗, or even A = A∗, for all objects A. In this talk, we investigate which coherence conditions should be required of such a category. We also investigate what graphical language could be used to reason about such a category. 1 Introduction 1.1 Self-duality It is well-known that each finite dimensional vector space A is isomorphic, but not naturally isomorphic, to its dual space A∗. Moreover, if A is an inner product space, then there exists a natural bijection between A and A∗, but (in the case of complex inner product spaces) it is skew linear instead of linear, and therefore again A and A∗ are not naturally isomorphic. On the other hand, there are autonomous categories (such as the category of finite dimensional real inner product spaces, or the category of finite sets and relations) that are equipped with a naturally arising family of isomorphisms A =∼ A∗. It therefore makes sense to study autonomous categories that are equipped with a family ∗ of isomorphisms hA : A ! A . Such a family may either arise naturally from the category itself, or it might be imposed by arbitrary choice as an additional structure. In any case, it makes sense to ask which coherence conditions, if any, the isomorphisms hA should satisfy. It also makes sense to consider a strict version in which A = A∗, and to ask whether any coherence conditions should be imposed. Finally, in light of the fact that there are sound and complete graphical languages for various notions of autonomous categories [6], it makes sense to ask whether there is a sound and complete graphical language for autonomous categories with the additional structure of A = A∗. A similar structure of self-duality was recently described, in the more general context of involutive monoidal categories, by Egger [1]. 152 Selinger 1.2 Self-duality without coherence There are two possible approaches to coherence. The first is to start with an autonomous (for example, compact closed) category, and to equip each object A with a chosen isomor- ∗ phism hA : A ! A , without requiring any naturality or coherence conditions at all. In this approach, each object is equipped with a chosen “self-duality structure”, which is in- dependent of any other object. It is akin, for example, to equipping each finite dimensional vector space with an arbitrarily and independently chosen basis. In this case, the structure chosen on, say, A ⊗ B or A∗ does not need to bear any relationship to the structure chosen on A or B. There is an obvious sound and complete graphical language, namely the usual language of autonomous categories, extended with basic boxes A A* hA and their inverses, satisfying no special laws. The strict version of this notion is to require A = A∗ for all objects without any coher- ence. Equivalently, for each object A, one requires a unit η^A : I ! A ⊗ A and a counit ^A : A ⊗ A ! I, satisfying the usual two laws for an exact pairing, i.e., idA⊗η^ η^⊗idA A A ⊗ A ⊗ A A A ⊗ A ⊗ A ^⊗idA idA⊗^ (1.1) idA idA A; A; with no additional conditions imposed. Note that, in the absence of coherence conditions, it is not possible to write A A η^A = and ^A = A A in the graphical language, because the graphical language would then validate additional laws (think of the graphical representation of η^A⊗B, for example). So even in the strict case, the only available graphical language (which is then sound and complete) is the same as in the non-strict case, i.e., A A h η^A = and ^A = A : (1.2) −1 hA A A In particular, even in the strict formulation there is no reason to expect any particular equations to hold between morphisms. For example, in the presence of a symmetric monoidal structure (or more generally, a pivotal structure), there is a canonical isomor- ∗∗ phism iA : A ! A , usually depicted as the identity in the graphical language. In the setting of strict self-duality without coherence, we of course have A = A∗∗; however, there ∗∗ is no reason to expect the canonical map iA : A ! A to be equal to the identity of A. ∼ Autonomous categories in which A = A∗ 153 Indeed, in the graphical language, the map iA will then have to be depicted as hA −1 hA A A; which is not a diagram for the identity morphism. 1.3 Self-duality with coherence In light of the above, it seems reasonable to require the self-duality isomorphisms hA : A ! A∗ to satisfy some coherence axioms. Preferably the coherence axioms should be chosen in such a way that (a) they are valid in relevant examples, and (b) there is a good graphical language. Note that the assumption A =∼ A∗ is not as benign as it might first appear. For example, because an isomorphism (A ⊗ B)∗ =∼ B∗ ⊗ A∗ is present in any autonomous category, we obtain a derived isomorphism 1 1 h ∼ h− ⊗h− A ⊗ B −−A−⊗−!B (A ⊗ B)∗ −=! B∗ ⊗ A∗ −−B−−−−A! B ⊗ A: (1.3) If the original category was assumed to be symmetric monoidal (or more generally, braided monoidal), one would certainly want to require (1.3) to be equal to the symmetry (or braid- ing). And even if the underlying category was not assumed to possess a braiding (or sym- metry), the self-duality assumption forces a braiding upon us via (1.3). The morphism (1.3) also has implications for the graphical language. The isomorphism (A ⊗ B)∗ =∼ B∗ ⊗ A∗ is usually represented in the graphical language like an identity (and in planar autonomous categories this is the only possible choice). Therefore, because of ∗ (1.3), it is not possible to represent hX : X ! X as an identity in the graphical language for an arbitrary object term X. As the case X = A ⊗ B shows, the map hX should instead be represented as a half-twist: A A* (1.4) The morphism (1.3) then becomes: B A = B A A B A B ∗ Also note that, if hA : A ! A is represented in the graphical language as a half-twist, then the morphism ∗ ∗ hA∗ ∗∗ hA ∗ θA∗ := A −−! A −−! A (1.5) is represented as a full twist on A∗: This indicates that an autonomous category with self-duality should at minimum be tortile (and possibly compact closed if the braiding is a symmetry). Or in other words, the minimal autonomous structure on which it makes sense to require a self-duality is that of a tortile category. 154 Selinger 2 Background We recall some well-known definitions from the theory of monoidal categories [4]. Definition (Braided monoidal category). A braiding on a monoidal category is a natural family of isomorphisms cA;B : A ⊗ B ! B ⊗ A, satisfying the two “hexagon axioms”: (idB ⊗ cA;C) ◦ αB;A;C ◦ (cA;B ⊗ idC ) = αB;C;A ◦ (cA;B⊗C) ◦ αA;B;C; ⊗ −1 ◦ ◦ −1 ⊗ ◦ −1 ◦ (idB cC;A) αB;A;C (cB;A idC ) = αB;C;A (cB⊗C;A) αA;B;C: A monoidal category that is equipped with a braiding is called a braided monoidal category. Definition (Balanced monoidal category). Recall that a twist on a braided monoidal cat- egory is a natural family of isomorphisms θA : A ! A, satisfying θI = idI and such the following commutes for all A; B: cA;B A ⊗ B B ⊗ A (2.1) θA⊗B θB ⊗θA A ⊗ B B ⊗ A: cB;A A balanced monoidal category is a braided monoidal category with twist. Definition (Right autonomous category). Recall that a right dual for an object A in a monoidal category is given by (B; η; ), where B is an object, and η : I ! B ⊗ A and : A ⊗ B ! I are morphisms, such that the following two adjunction triangles commute: idA⊗η η⊗idB A A ⊗ B ⊗ A B B ⊗ A ⊗ B ⊗idA idB ⊗ (2.2) idA idB A; B: The maps η and determine each other uniquely, and moreover, the triple (B; η; ), if it exists, is uniquely determined by A up to isomorphism. A monoidal category is right ∗ autonomous if every object A has a right dual, which we then denote (A ; ηA; A). In the graphical language [6, 3, 2], these structures are represented as follows: Braiding cA;B = ; Twist θA = ; A A* Dual ηA = ; A = : A* A In each case (braided, balanced, or right autonomous) there is a coherence theorem stating that an equation follows from the respective axioms if and only if it holds in the graphical language. When combining a balanced structure with an autonomous structure, an additional ax- iom is required that relates them. In the presence of this axiom, coherence holds for the combined graphical language. ∼ Autonomous categories in which A = A∗ 155 Definition (Tortile category). A tortile category is a balanced monoidal category which is also right autonomous and satisfies ∗ θA∗ = (θA) : A compact closed category [5] is a special case of a tortile category satisfying θA = idA −1 for all A (and therefore cA;B = cB;A). Equivalently, a compact closed category [5] is a right autonomous symmetric monoidal category. 3 Self-duality structure on tortile categories ∗ We will state the coherence axioms for the self-duality isomorphisms hA : A ! A in two different ways. The first formulation, given in this section, assumes that the underlying category is tortile (or compact closed, which is a special case).
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